Two examples of this variational approach

As an example of the use of this method, consider a reaction between two species, as usual with a boundary condition at infinity where p - 1. Let the reactive sink term be given by the Heaviside step function / — u(R — [rj). The solution for the rate coefficient has already been given (Chap. 8, Sect. 2.4) as 

Doi [485] proved that the trial function, p, which maximised /o[ p], was indeed the density found from the exact solution of the diffusion equation. He then suggested that, by way of an example, a trial function of the form = Arn could be used. Now 

Furthermore, integration of the Green s function over alt time gives 

The maximum value of k2 is where (kR2jD) = 3n(2n + 5)2( + 3)/ (2/7 + 3)2. For large values of k, the lower bound reverts to the Smo-luchowski limit 47tJRjD. However, optimising the value of n gives a result within 1% of the exact answer above. 

An interesting extension of this example is to consider two stationary sinks located at rt and r2, each with a sink function given by the Heaviside expressions 

---